# Embedding problems for automorphism groups of field extensions

## TL;DR

This paper proves that finite split embedding problems over any field can be regularly solved without requiring the solution fields to be normal, advancing inverse Galois theory.

## Contribution

It provides an unconditional proof for a key case of the conjecture, extending prior results on automorphism groups of field extensions.

## Key findings

- Embedding problems can be solved without normality constraints
- Extends previous automorphism group realization results
- Progress towards the inverse Galois conjecture

## Abstract

A central conjecture in inverse Galois theory, proposed by D\`{e}bes and Deschamps, asserts that every finite split embedding problem over an arbitrary field can be regularly solved. We give an unconditional proof of a consequence of this conjecture, namely that such embedding problems can be regularly solved if one waives the requirement that the solution fields are normal. This extends previous results of M. Fried, Takahashi, Deschamps, and the last two authors concerning the realization of finite groups as automorphism groups of field extensions.